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arXiv · 1310.8313

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

Flavia Bonomo·Oliver Schaudt·Maya Stein·Mario Valencia-Pabon
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Abstract

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χ_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ(G),\ldots,χ_b(G), and b-monotonic if χ_b(H_1) \geq χ_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.

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Flavia Bonomo, Oliver Schaudt, Maya Stein, Mario Valencia-Pabon. 2014-01-10. b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs. https://doi.org/10.1007/s00453-014-9921-5

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